The complexity of classical music networks Vitor Guerra Rolla - - PowerPoint PPT Presentation

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the complexity of classical music networks
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The complexity of classical music networks Vitor Guerra Rolla - - PowerPoint PPT Presentation

Aug 06, 2023 148 likes •520 views

The complexity of classical music networks Vitor Guerra Rolla Postdoctoral Fellow at Visgraf Juliano Kestenberg PhD candidate at UFRJ Luiz Velho Principal Investigator at Visgraf Summary Introduction Related Work Musical Networks

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The complexity of classical music networks

Juliano Kestenberg

PhD candidate at UFRJ

Luiz Velho

Principal Investigator at Visgraf

Vitor Guerra Rolla

Postdoctoral Fellow at Visgraf

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Summary

Introduction Related Work Musical Networks Scale-free Small-world Results Fractal Nature of Music Conclusions and Future Work

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Introduction

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Introduction

40 pieces of classical music → MIDI format Bach (6), Beethoven (9), Brahms (1), Chopin (1), Clementi (6), Haydn (5), Mozart (7), Schubert (4), and Shostakovitch (1) Built a network from each piece of music Perform scale-free and small-world tests

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Related Work → Music

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Related Work → Music

  • Liu et al.

“Complex network structure of musical compositions: Algorithmic generation of appealing music” Physica A: Statistical Mechanics and its Applications (2010)

  • Perkins et al.

“A scaling law for random walks on networks” Nature Communications (2014)

  • Ferretti

"On the Complex Network Structure of Musical Pieces: Analysis of Some Use Cases from Different Music Genres" Multimedia Tools and Applications (2017)

I.F. 12,124 18 citations I.F. 2,243 63 citations I.F. 1,530 1 citation

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Related Work → Music

  • Liu et al.

“Complex network structure of musical compositions: Algorithmic generation of appealing music” Physica A: Statistical Mechanics and its Applications (2010)

  • Perkins et al.

“A scaling law for random walks on networks” Nature Communications (2014)

  • Ferretti

"On the Complex Network Structure of Musical Pieces: Analysis of Some Use Cases from Different Music Genres" Multimedia Tools and Applications (2017)

I.F. 12,124 18 citations I.F. 2,243 63 citations I.F. 1,530 1 citation Scale-free: Yes Small-world: Yes Scale-free: Yes Small-world: No report Scale-free: Yes Small-world: Yes

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Related Work → Music

  • Liu et al.

“Complex network structure of musical compositions: Algorithmic generation of appealing music” Physica A: Statistical Mechanics and its Applications (2010)

  • Perkins et al.

“A scaling law for random walks on networks” Nature Communications (2014)

  • Ferretti

"On the Complex Network Structure of Musical Pieces: Analysis of Some Use Cases from Different Music Genres" Multimedia Tools and Applications (2017)

I.F. 12,124 18 citations I.F. 2,243 63 citations I.F. 1,530 1 citation 202 pieces → Classic & Chinese Pop 8473 pieces → Folk from Europe & China 8 pieces → Rock, Blues, Jazz...

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Related Work → Math Tests

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Related Work → Math Tests

  • Clauset et al.

“Power-law distributions in empirical data” Siam Review (2010)

  • Watts & Strogatz

“Collective dynamics of ‘small-world’ networks” Nature (1998)

  • Newman & Watts

"Renormalization group analysis of the small-world network model" Physics Letters A - Elsevier (1999)

I.F. 40,137 35731 citations I.F. 4,897 5947 citations I.F. 1,772 1364 citations

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Musical Networks

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Musical Networks

Mozart’s Sonata No. 16 (KV 545) first bar

(d)

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Musical Networks

Project's website: http://w3.impa.br/~vitorgr/CNA/index.html Python/NetworkX Software for complex networks https://networkx.github.io/

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Scale-free Property

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Scale-free Property

Node degree distribution → Power law estimation Least squares method (Old)→ used by Liu and Perkins Clauset's test

  • Cohen & Havlin

"Scale-free networks are ultrasmall" Physical Review Letters (2003) I.F. 8,462 801 citations

2 < α < 3

  • i. Maximum likelihood estimation (α)
  • ii. Kolmogorov-Smirnov ( p-value > 0.1 )
  • iii. Likelihood Ratio (LR)

power law vs. alternative hypotheses:

log-normal, exponential, stretched exp

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Related Work → Music

  • Liu et al.

“Complex network structure of musical compositions: Algorithmic generation of appealing music” Physica A: Statistical Mechanics and its Applications (2010)

  • Perkins et al.

“A scaling law for random walks on networks” Nature Communications (2014)

  • Ferretti

"On the Complex Network Structure of Musical Pieces: Analysis of Some Use Cases from Different Music Genres" Multimedia Tools and Applications (2017)

1 < α < 2

1,05 < α < 1,28

No report

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Small-world Property

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Small-world Property

Mean Shortest Path Length (MSPL) Six degrees of separation – Myth Average Cluster Coefficient (ACC) Fraction of triangles Musical Networks vs

Random Networks & Small-world Networks

(near equivalent)

Newman, Watts Strogatz

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Related Work → Music

  • Liu et al.

“Complex network structure of musical compositions: Algorithmic generation of appealing music” Physica A: Statistical Mechanics and its Applications (2010)

  • Perkins et al.

“A scaling law for random walks on networks” Nature Communications (2014)

  • Ferretti

"On the Complex Network Structure of Musical Pieces: Analysis of Some Use Cases from Different Music Genres" Multimedia Tools and Applications (2017)

No report

Small-world !!!

No report

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Results → Scale-free

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Results→Scale-free

Clauset’s test – i & ii steps:

(a) Sonata No. 23 in F minor (Appassionata) Opus 57 (1804) composed by Beethoven, (b) Sonata No. 12 in F major KV 332 (1783) composed by Mozart, (c) Piano Sonata in D major Hoboken XVI:33 (1778) composed by Haydn, (d) Violin partita No. 2 in D minor BWV 1004 (1720) composed by Bach, (e) Sonatina in F major Opus 36 No. 4 Opus 36 (1797) composed by Clementi, and (f) Sonatina in C major Opus 36 No. 3 Opus 36 (1797) also composed by Clementi.

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Results→Scale-free

Clauset’s test – iii step:

(a), (b), and (c) present the scale-free property. (d) behaves more like a log-normal (e) behaves like an exponential distribution (f) did not behave like any distribution tested.

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Results → Small-world

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Results→Small-world

MSPL and ACC for musical networks, random networks, and small- world networks.

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Final → Results

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Results

Scale-free Not Scale-free

52,5%

Small-world Not Small-world

B a c h B e e t h

  • v

e n B r a h ms C h

  • p

i n C l e me n t i H a y d n Mo z a r t S c h u b e r t S h

  • s

t a k

  • v

i c h B a c h B e e t h

  • v

e n B r a h ms C h

  • p

i n C l e me n t i H a y d n Mo z a r t S c h u b e r t S h

  • s

t a k

  • v

i c h

62,5%

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Fractal Nature of Music

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Fractal Nature of Music

  • Schroeder

“Is there such a thing as fractal music?” Nature (1987)

I.F. 40,137 19 citations

  • Henderson-Sellers & Cooper

“Has classical music a fractal nature?—A reanalysis” Computers and the Humanities (1993)

I.F. 0,738 10 citations

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Fractal Nature of Music

Fractal Dimensioning vs. Complex Network Analysis

  • Song et al.

“Self-similarity of complex networks” Nature (2005)

I.F. 40,137 1102 citations Self-similarity Mandelbrot

Scale-free property

Newman

  • Song et al.

“Origins of fractality in the growth of complex networks” Nature Physics (2006)

I.F. 22,806 424 citations

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Conclusions & Future Work

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Conclusions

Previous work (Liu et al., Perkins et al., Ferreti) disregarded:

– Harmony – One piece per network – Updated statistical methods → Clauset et. al.

Our work suggests that classical music may or may not present the scale-free and the small-world properties

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Future Work

Evaluation of other music genres Investigation of edge weight distribution Evaluation of fractal dimension according to Song et al. algorithms Understanding the community structure of our musical networks.

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Computer Music @ VISGRAF

Thank you!

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Extra – Hubs

Although we provide a precise evaluation of the power law, our musical networks did not present a long tail as many scale-free networks, i.e., we could not identify a small number of nodes with very high degree. On the other hand, according to Janssen due to the finite size of real-world networks the power law inevitably has a cut-off at some maximum degree. Such a cut-

  • ff can be clearly verified in Figures 2(a), 2(b), and 2(c).
  • Janssen

"Giant component sizes in scale-free networks with power-law degrees and cutoffs" Europhysics Letters (2016)

I.F. 1,957 3 citations

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Extra – ACC

Local clustering coefficient for undirected graphs: Average cluster coefficient:

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Extra – Cohen & Havlin

  • Cohen & Havlin

"Scale-free networks are ultrasmall" Physical Review Letters (2003) A power law distribution only has a well-defined mean over x [ 1 , ∞ ], ∈ if a > 2. When a > 3, it has a finite variance that diverges with the upper integration limit

I.F. 8,462 801 citations

2 < α < 3

∫

xmax as 〈 x

2

〉 = xmax xmin x

2

P ( x ) ~ xmax

3-a